Heat Transfer Calorimetry

7/29/2019 Heat Transfer Calorimetry

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cpp (Calorimetry &Thermal ElCpansion) 6th March It

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PART. I : SUBJ ECTIVE. QUESTIONS

S,~jFJJ'J~~!.-.sL I M I'T.E'D'

Section (A) ,:Calorimetry

A 1. ' ,In the following equation calculate the value of H.1 kg steam at ZOO.C=H +1 kg water at 100.C (5.-=C onstanl = 0.5 Callgm.C)

A 2.' Fromwhat heightsh~Uld a piece of ice-(O.C )full so that itmelts complelely? Only one-quarter of the energyproduced is absorbed by the ice as heat. (Latent heal of ice =3.4 x l()5J kg-', g.= 10m/s2)

A 3. A copper cube of mass 200g slides down on a rough inclined plane of inclination 37. at a constant speed.

Assume that any loss in mechanical energy goes into the copperblock as thermal energy. F indthe increase

in the temperalur~ of the block as it slide~down 60 em. Specific heat capacity of coP ,:~:=\20J /kg.K

A 4. A paddle wheel is connected with a block of mass 10 kg as shown In figure. ~,

The wheel is completely immersed in liquid of heat capacity4000 J IK . The (- ,if/\container is adiabatic. For the time interval in which block ~oes down;~t m ,,"U/sloWlycalculate ( ',eo ,' \,::..,.A(a) Work done on the liquid '. - , ,.'

(b) H eat s upplied to the liquid ' ~:,(\", t(c) R ise in the temperature of the liquid . ", " \Neglect the heat capacity of the container and the paddle. (g =10 nillf) "

Seetron (B) : Tlierm&lExpa~SI6~ > \ j~~ . .',.,. .B 1. :"he temp~rature of a m~tal ban!s ~~;Arrange the ~reenla9cechange in volume, surface area and radius

InascendIngorder. ~ '\ ' " \ " \ ,'; p '_B 2. A brass disc-fits in a'holein\a steel plate. Would youh'eal or cool the system 10 loosen the disc from

the hOle?A-i''nte lh~t\ci.


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PART. II : OBJ ECTIVE QUESTIONS'

Marked Questions are, having more tllan one correct optlo'n.

SectIon (A): CalorImetry

A 1. Asr.lall quanlily, mass m, of water-at a temperaluree (inOC)is poured on loa'iarge mass M of ice which is

al its mel~ng point If c is the specific heat capacity of waler and lthe lalent heat of fusion of ice. then themass of ice melled is given by'; ,

(A) Mlmc6

(8) mceML

(C) Mc6L

(D) mceL

A 2." When m gm of waler at 10'C is mixed with m gm of ice at O'C, which of the following slatements are'

false? ' ' , t it ''' '(A) The temperature. of Ihe syslem will be given by the equation ' ..', " '. ; " . " , ) l!>

m x 80 +m x 1 x (T - 0) = m x 1 x (10 - T) ".,J!lfp ,(B) Whole of ~ce w~1Imelt and temperature w~1Ibe more than O'C b.utlesser tha,~;10'C Jl(C) Whole of Ice Will melt and temperature Will be O 'C . t; ',.,(0)Whole of ice will not melt and temperature will be O 'C . \

SectIon (8) : Thermal expansion .' 'G, ,' ~~/~',B 1. Two large holes are cut in a metal sheetlfthls'is heated, distancesABand BC, (as shown)

, ....~ \~t:!\,~~B~)t...ic~~~;;:r~,"\.~.~~~~;::"~_

"1# ~ Y \ .J~ 'B 2. A-steel scale is to be prepared'such thaUhe millimeter intervals are to be accurate within 6 x 10" mm.

Th~'.maxi';;um temperatute variation from the temperature of calibration during the reading of the millimeterarksJ s (a~ 12 x 1 0 - < I*C)"Y

;.{(A)i~~'\ (B) h'c (C) 5.0'C ' (D) 5.5'CEx~Sion duringljeating - I

. (A j"occurs only in a solid , (B ) increases the density of tile material~.. ~ ' _"0, '"

(C) decreases the density of the material (D) occurs at the same rate for all liquids lind solids.\IV " ,

B 4. If a bimetallic strip Is heated, it will

(A) bend towards the metal with lower thennal expansion coefficient.

, (B) bend towards the melal with higher thennal expansion coefficient.

(C ) twisl itself into helix. -

(D) have no bending,

B 5*. Two identical beakers with negligible Ihennal expansion are fitted with water 10 the same level at 4'C.

If one says A is heated while the other says 8 is cooled. then:

(A ) water level in A must rise (8) waler level in B must rise

(C ) water level in A must fall (D) water level in 8 must fall

Section (C) : Temperature

C 1. A difference of temperature of25' C is equivalentto a difference of :(A)45' F (8)72.F (C)32F (D) 25 F

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EXERCISE#2

PART - I:SUBJECTIVE QUESTIONS

A thermally Isolated vessel contain~ 100 9of water at OOC .When air above the water is pumped out, someof the water freezes and some evaporates at OO Citself. Calculate the mass of the ice formed if nO,water is fell

in the vessel. latent heat of Y aJ lC!rizationof water at WC = 2.10 x 10" J lkg and latent heat of fusion ofice = 3.36 x 10" J lkg.

A pitcher oontalns 20 kgof~ter. 0.5 gm of water comes out on the surface of the ' 0 . , . ',pitcher every second through the pores and gets evaporated taking energy from the ,

remaining water. C alculate the approximate time in which temperature of the water

decreases by 5"C :Neglect backward heat transfer from the atmosphere to the water.

Specific heat capacity of water = 42ooJ lKgoC. ,latent heat of vaporization of water 2.27 : .10"J lKg ',' A~

A thermally insulated, closed copper vessel.conlains water at 15OC.W hen the vessel is shaken vigorously

for 15 minutes, the temperature rises to 17.C . The mass of the vessel is 1CO gand that"Oftll8'Watefis 200g.~. . ,.,:,." .. , . .:.4f3a. and h = h, then calculate, the volume of liquid overflow, ,(e) If the surface of a cylindrical container is marked with numbers for the measurement of liquid level of

liquid filled inside itAss uming correct marking at inilia!' temperature if we increase the temperature

of the system by 46 then .

(i),' F ind height of liquid level as shown by the scale on the vessel. Neglect expansion of liquid

(Ii) Find height of liquid level as shown by the scale on the vessel~'Neglect expansion ofcontainer.

(iii) Find relation between TLand llc so that height of liquid level with respec-l to ground

(1) increases (2) decreases (3) remains constant.

,.

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6. A metal piece weighing 15g is heated to 100.C and then immersed in a mixture of ice and water at the

thermal equilibrium. The volume of the mixture is found to be reduced by 0.1S em" with the temPerature of

mixture remaining constant. Find the specific heat of the metal. Given specific graVity of ice " 0.92, latent.heat of fusion of ice = 80 callgm.

'7- 20 gm ice at-10.C is mixed with m gm steam at 100 .C.Find the minimum valut' ofm so thalfinal1y all ice.

and steam converts into waler. (Use s=O.S callgm.C,s_ =1 cal/gm.C.L (melting)=80 callgm and L(vaporization)=540 callgm)

8. . A simple seconds pendulum is constructed out of a very thin string of thermal coefficient of linear expansiona" 20 x 1(t4/"C and a heavy particle attached to one end. The free end of the siring Is suspended from the

ceiling of an elevator at rest. The pendulum keeps correct time at DOC:When me temperature rises 10SOO C,

the elevator operator of mass 60kg being a studenl of Physics accelerates the elevator vertically, to havethe

pendulum correct time. F ind the apparent weighl of the operator When the pendulum keeps correct time at

SO"C. (Take g = 10 m/s')

PART. II : OBJECTIVE QUI;STIONS

Singiechoice type ....~ ., . . ~.

1. A melal ball of specific gravity 4.5 and specific heat 0.1 callgmp"C is placed on a larg~sl~2~.~tlCtl al

O.C. Half of the ball sinks in the ice. The inilial temperalure of !he balli:~ "--

-41~'1rAlf),(Latenl heal capacity of ice = 80 cal/g, specific ~.r;lIy,ifY\ofice "'.0,9 ./ ~(A)100.C' . (B)90.C {~)80.C" {;(Q)J O'~G' . .

. ,~. \ l' 0 ; .. . " ; A .

2. . A sleel rod.25 cm long has a cross-sectiqnal area of 0,8 cl'l1" Force. requiredtostfetch this rod by.the. 'same amounl as the expansion produCed-by heatingitthro'ligh 10.Cis:~ . . .

.-J '- -'''2---=",'''' '~ \.. ..~., .' ~(Coefficient of Iinearexpansicln ()f,steelis 10""C and Y ourig'sinodulus of steel is 2 x 10'N/rTi'.) .

(AI160 N . /~~)36~,..t\ (C) \06)r'~' (D) 260N. .. 3. If I is the moment of inertia of a solid.body having a -coefficient of linear explinslon then Ihe change in,- :~""'.~:''. ~:'~~,-~,.....-. . .

I corresponding to asniall change in temperature IITis

(~laIll~i . \;ri~lllT' \!A cl2aIllT . (Of3aUT .

4.;* AJ oiq)idWith qoeffiCie!11of ~Ii~~ansion y is filled in a container of s.material having the coefficient of." .linearexpansiona .lIthe IiquiCloverflows on heating,then-

, ,~ A ffly \,P(Blr


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More th'an one choice type

8. When two nonreactive samples atdiffe~nttemperatures are mixed In an isolated'containerotnegligibleheat capacity the temperature of the mixture can be: .' . .

(A) lesser than lower or greater than higher temperature'

(8) equal to fower or higher temperature

(Cl greater than lower but lesser than higher temperature

(D) average of lower and higher temperatures

9._ There is a rectangular metal plate in which two cavities in the shape of rectangle and circle are made,as shown with dimensions. P and aare the centres of these cavities. On healing the plate, which ofthe foliowing quantities increase?

(C) Height of cylinder outside the liquid remairlS constant. .'

(A) volume of cylinder inside the liquid remains constant'\:,.-- .

(B) volume of cylinder outside the liquid remains constant .

(A) nr2. , (8) ab

+0-f p

Column II

(PlY =0

(q)y =2a

d(rly= 3a p

(0) Height of cylinder irlSide the liquid remain constant . d(s)y = ( 2a +a - )p

In the following question column - I represents some physical quantities & column-ll represents their units,match them

Column I Column II

(A) . Coefficient of linear expansion (p) CaIrC .,(B) Water equivalent (q) gm

(C) heat capacity (r) (OCr'

(D) Specific heat (5) CaUgOC

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PART ~\I : COMPREHENSION

Comprehension #1 "

A 0.60 kg sample of water and a sample of ice are placed in two compartments A and B that are

, separated by a conducting wall, i,na thermally insulated container. The rate of heat transfer from the

water to the ice though the conducting wall is constant p. until thermal equilibrium is reached. The

temperature T of the liquid water and the ice are given in graph as functions of time l. Temperature of

the compartments remain homogeneous during whole heat transfer process.

Given specific heat of ice =2100 jlkg-K

Given specific heat of water =4200 J /kg-K

Latent heat of fusion of fce= 3.3 x 10sJ /kg

(0)100.C

20

(C) 94.C

-- .-----~-..' .------- B',------- ..' ....-.water--- .Ice -- ----...... '.-------f'Z. .

______ ~.. ,,' ,t .

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...Insulatingconducting wali -20

Th'~I~~rn":4,: ~(A) 42.0 W -"'. (B).~6'~""fi \. (C) ~J ~;~The initial mnss oHhe ICeIn.thecontalner IS equal to, '

(A) 0.36 ~ \ (~( 1.2 kg'~~I!\(Cl'2,:i


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_.ExerCise#4_PART. I : liT..lEE PROBLEMS (PREVIOUS YEARS)

[J EE .2003 (SC r.).3184,-1J

.!(A) (B) & S(C~ \ \. ( t . .

~m_~ ;7~,''\,\~~'-l7~1UPPedWhen a block of iron f1oatsJ nmercury.afO.C a fraction kidf its volume is submerged, while at the

,_ . " . , .. , ;C=-, .-"\ \-, ,.:0

temperature 50.C, a fraelion'!. iSseeil lobe submerged. If lJ J ecoefficient of volume expanSlOnof Iron

~"\ '"",,/. ~.k'>(~~;1. . . .

is TFei(lha~ ofm~~7}\.'then~~ ~li\ k; ca~ be _expressed as:

.'.M~.?\\':1~F'

Vf+6OyF.' [J E~::::~scr.), 1/35,-1)

.. (e;~{OTH~,1ij~\ (Bh\t\lOiHv. (} 1-6OyHv (D) 1+6OyFe '. .

An lca'c\lbe of'mass 0,1 kg at O'C is placed In an isolated container which Is at 227'C. The specific

"j'lleal S of the con\alnlir varies with temperature T according to the empirical relation S = A + BT, where"A" l'OO.cal/kg-Kand B =2 x 1lt" callkg-K" . If the finalteinperature of the containerls 27.C, Determine

._ " " " " '"'the mas~oflhe conlainer. (Lalent heatoffusion of water=8 x 10' callkg, sp. heat of waterc10' cal/kg-I


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2 liters water at 27"C is heai~d by~1 kIN heai~r 'i n a n open container. On an verage heat is lostlosurroundings at the rate 160 J /s. The time required for the temperature to reachn"C is .,

. " {J EE-200S (Scr.l,3/t!4. ~1J(A) 8 min 20 sec (8) 10min (e)7 min (0) 14min

In an insulatedvessel, 0.05 kg steam at 373 K ,and0.45 kg of ice at 253 K are mixed. Find:lIle final

temperatureof the mixture (in Kelvin).. {J EE 2006, 61184.-1]

Given,L""", =80c


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Answers

PART.III

9 . (A) 10. (C) 11. (A)

PART-IV

12- (I) True (Ii) True . (iii) True(Iv) True

I PART.V

13. solid sphere'

(ii) hollow sphere = solid sRhere

Section (e): ~ .~

C 1. (a) All tie (ti)50~X.50OY.50.W;

\~,if


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M G 8':PA RT - I:OBJECTIVE QUESTIONS

'.

I ' C

Sle~m at 100' C is 'passed'into 1.1 kg of water contained In a calorimeter of water equivalent O.02 kg at15'C till the temperatUre of the calorimeter and its contents rises to S O'C . The mass of the steamcondensed in kilogram is : [JEE '8B, 2)(A) 0,130 (8) 0.065 (C) 0.260 (0) 0.135

A piece of metal floats on mercury. The coefficient of volume expansion of the metal and mercury are

1, & 12 respectively . If the temperatilres of bOth mercury and the metal are increased by an amount ,

AT , the fraction of the volume of the metal submerged in mercury changes by the factor _ ' __

(Ratio of final fraction to the initial fraction) pEE '81,2)

Two vertical glass tubes filled with a liquid are connected a.ttheir lower / r '& . . ' _ends by,a horizontal capillary tube. One tube Is surrounded by a bath i .' r n r :C? ntaining }ce and water atO.~and ~e other by hot water at t.C . The I : ~ ~dIfference In the height of the liquid In the two colum~ Is All, and the ',II :height of the column at O.C is h". Coefficient of volume expansionof ' J t t l D :the liquid 1I I l :

, AI.A gas thermometer is used as a standard thermometer fOrmeasurJ mentof temperature. When the

. ~ ~ .., ',-". ~. It ).

gas container of.lhe thermometer Islmniersed in water at ils triple point 273.16 K; the pressure In theA' ~ --- '~ ,~ ~-- ~

gas thermometer reads 3 x 1O' WiTt': WIleilthe gas container of the same thermometer Is Immersed in"another system, the gas preuuTe7e2cls3.5 x 10' N/~emperature of this system is therefore. ,

"__ 'C. A, ~ ~, [JEE ,87, 2J .Earth receiv~.s 140Q}'-"/~of solar 'p,ower ..If all the so.lar energy faliing on a lens of area 0.2102 isfocused on'to a block:pf ice of mass 280 g~niS, the time taken to melt the ice will be minutes.(Latentlieat affusion ofice-:~.3 x 105''J ~. [JEE '87, 2]

,. .P A .SUBJE:CTlVE QUESTIONS

A 50~"tead 'b~liet. ;sp,eclfic.heat 0.02 callgm Is mllially qt'30' C . It is fired vertically uPr'3rds with a~peed .0 J 840m/sec &'9n returning to the starting level strikes a cake of ice at 00C . How much ice ismelted. Assumi;'that all energy is spent in m~lting only.[Latent heat of ice = SOcallgm J '[REE ~8, 6.J'The temperature of 1OOgm of water is to be raised from 24. C to 90'C by adding steam to it~CalC ulatethe mass of the steam required for this purpose.', [J EE '9B,21 '

An":Iectricai heating coil was placed In a calorimeter containing 360 gm of water at 100C '. The coilconsumes energy at the rate of 90 watt. The water equivalent of the calorimeter and the co" is 40 gm.Calculate what will be the temperature of water after 10 minutes, [REE '85, 7].J = 4.2 J oules/cal. 'One gram of water (volume = 1 cm') becomes 1671 em' of steam when boiled at a pressure of one

atmosphere. Latent heat of vaporization afthis pressure Is 539 callgm . Compute the worK done.'[ 1 atm = 1.013 x 10' Nrn-2j ,{REE'8B,3)

A sleal rod 25 em long has a cross-sectional area of O.Scm2 What force would be required to stretchthis rod by the same'amount as the expansion produced by heating it through 10.C? (Coefficient oflinear expansion of steel Is 10"/"C and Young's modulus of steel is - 2 x 10" N/m2.) [J EE '89,3]

The hrass scale 0'a barometer gives correcl reading at O.C . Coefficient of thermal expansion of brassIs O .lIOOO2J OC. The barometer reads i5 ern at 27" C. What is the currecl atmospheric pressure at .27.C~' '[ J EE 'B9. 21

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.'7. A clock with an iroi, penilulum keeps co,rrecttime at 20. C. How much will It lose or gain in a day if the

tem~:erature changes to 40' C? (Coefficient of cubicai expansion of iron = 0.000036FC ) I J EE '80,3,1

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8. Two r~~. of different.'TIelals h.avingsamearea~f cross section A are placed end to end betwee~~m?sslVe platforms, as shown In thefigure.T~e.f,rstro~~asa length l" coefficient of.liriear ex ails' ,.'

.aland Y oung'lrmodUIUs V,. ~he cor~espondlO9qu;mlitl~s for the seco~d rod are L.,.a,;.and~: :fD,ntemperallJ re of both the rods.ls nowIncreased.byT"C. Find the force with which the rods act 0 ~,'..,~eother ( at the higher temperature) Intermsof given quantities. Also find the lengths of the rOdsna~-:;;hhigher temperature. Assume lhat thereisnochangeIn lhe cross sectional area of the rods and that th:

rods do nol bend. Thete is no deformationof lhe walls. (JEE 'so, s 1

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11.

12.

14.

15.

A compos~e rod is made by joining a copperrod end to end with a second rod of different,material butof the same Ctoss section. At 25'C lhecompositerod is 1 m in length of which the length~~fthe COpperrod is 30cm.At 125'C the length of thecomposHerod Increases by 1.91 mm. When tI\ecoliiiiOsile rod'is not allowed to expand by holding Hbetweentwo rigid walls ilis found that the length~of.the twoconst~uenls do not change with the rise of temperature: Find the ~~!ts mOdGIUS :~rl!ih'e'iinear

expan~ion of ~he second rod given thai Young's modulus of.for~, PP Elr=, 1.3,x 1011

,N,Im~ahd,thecoeffiCient ofllnear expansion of copper= 1.7 x 10"rC. ~ ..., . (JEE'SO, 4l

A piece of metal weighs 46 9 In air. When~ Is immersed In a IiqJ jdo!.sp.e~gravity1.~4 at 27!C'itweighs 30 g. When the temperature of liquidis rai!l~~2'C the metal J :!i~cew~ghS 30.5g:Specific

gravity 01Iiqllid at42'C is 1,20. Calculatethe coeff~e=~inear\e~a,::s:r~TOflhlj metal. [JEE 'Sl, 3]

A one !Her flask contains some mercurY .':l!is found that at differenl'tEl.mperallJ ~Els.thevolume of air .inside the flask remains. the sam--e;Wtiat!sthe VOlume-~cury in tlie'f1as,k? Coefficient of linear:expansion of glass = 9 x 1O"~~C!Coefficient ofvolume;:expansion of mercury =1.8x 10" ,. C.

TwoAluminium rod~tee3s.s-secti.~ equal length .'.( JEE 'S1, 3]

t.arejoined'rigidly sidebY sid~as shown'!"figure:.lnltiallythe rods are atwe. Find . .Aluminiumthe tenQlha f !hI !. rod al'ttieteltlperature O W yom;9:smodulus of elas~cily of the Steelaliln:tiniumanc stee! are Yrailc y'lrespectivlily'iJ id coefficient 01linear expansion Aluminium

of aluminiutn 'ana,steel ar: 'tand~ respectively.-

~".~ . . . . ,The attemate diSCSof ironandiitrb'tn. havingsamearea of cross-5ection , are cemented togetherto make

a.;..cyfn~~r ~n~t:1~ose iempe~ure coe~cient of.res!s~vityis zero. If th~change in temperature in~~male dISC S~\!1esame, determinetheratio~elr thickness and the ratiOof heat.produced inthem.crheresistivity of iron andcarbon at20"C are 1 x 10 and3 10-

5Q-m and their temperature coefficientof

'~siSti'nc;e are 5 x 10--'and _ 7.5 ;'10.' per'C, respectively.Neglect thermal expansion. tRE E 1988)

4kfir a melal scale of length 30 emerd anobject The scale is cafibrated for temp 20'C:(a) What Is the aetuallength of divisionwhich is shown as 1 cm by scale at40OC.

Given a. =2.10-' re.(b) What will be the reading of scale at40'C ifthe actual length of object is 10 cm.(c) What will be the actual length of objectat;40 'C if its measured length is 10 em.(d) What is %error In measurementfor part (b) and (c) .

. (e) If the linear expansion coeffiCientof object is a.=

4 10-' and neglecting the expansion Ofscalethen an~rs of (b) and (c) parts. .(~ If a. = 4 x 10-' and a. = 2 10-' then find answers.of (b) and (c) part.

The apparatus shown in the figure consists of lour glass columns 'connected by horizontal sections. The height of two central columns B & c : are 49 cm each. The two outer columnS A &D are open to th~

atmosphere. A & Care mainlained at a temperature of 95'C While the columns B&D are maintainedat 5'C ..The height of the liquid inA & D mea~ured fromth~ base line are 52.8 em & 51 em respectively.Determme the coefficient of thermal expanSion olthe liquId. (JEE '97.~!

A

OS '

c 095- 5'

'-.1

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_'----An-s-w-er-s-_

~. (A) 2 . 3 .Ahy=-

hot

5 . 5.5 min = 330 sec..

:

PART-II

.,

42.14"C75.0405 em

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129m

1.6x1Q2N. 2 .

5 .

52.875gm.169.171J10.3685

A T{L t, +l2a2)Y 'Y 2 .' l + L 1l2T{Y 1a,pYia2)F = l Y +l v length. of the first rod= , l v +J,,,v ,

, 2 2', ,'2 . "

t\=10{1+20 "4" 10-5}., .

t; =10{1- 20" 4 " 10-5}(~ .I,=10{1+40" 10.5}

.t.=10{1-40" .10.'}

. y=2"10-4'"C.'

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MOBSolutionPART .1

\

o'c

IO' ~c

gy!liven bythe sun = 14C'0 0.2 tEmr!Jy~rr;qu!rr;d 1,3rneilihe ice" 260 10'" 3.3. 1Q5\~... .."

1~.00'x0.2 t" 280 " 10~'" 3.3 10' .

t =330 s~r..

f{ 1+Y2aT

. . ~ = 1+V1AT

Inequilibrium, ..Pressure atA =Pressure:al B

h-t-pg.h) Pogo 0 . 0 1+yl

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1. Heal released by sleam =heal gained bywaler and calorimelry

ml+m"S (100-80)=(1.1+ 0.02)"S.(80-15)

m.540+20m=1.12 "65

m=0.130 kg.

2. Forfloating condition

mg =BV dog.=vd,g'where do=density of object

d, =densily of liquid

fractionof volume submerged in liquid(f,) = v;

do

f,= d ; : -

after increasing lemp, byaT

P~RT.1l

1. Heal released by bulleI =heal gained byice1 .

.~ -mv'+ m SAG=ml2 I

.! .. (50" 10-') (840)' " - .:! ..-+50 0.02 (30 - 0) = m 802 . U .

rn=52.875 gm

2 . Heatreleasedby steam =Heal absorbe~bywaterrn,L+ 111," S (100-90) =m, "S(90-24)

540 rn,+ 10m, = 66 m,

'66"100~ ;.1,=' 550- "12r;:m

-'

EE Ltd., 48, Gurul,ripa Complex, M.!'. Nagar, Zoue-II, Bhopal (M.P.), Ph.: 4253355,4253455 14


15/35

I, . ,,'.

3 .Energy supplied by coil =heat gain by water and calorimetry=> Pt = (m, +w) " 8 A9=> 90 x (10 " 60) =(360 + 40) x 4.2 x ( C l - 10)

9=42.14.C .

I

. (1)

P rocess is isobaricw = P(V -V) = 1.013" 10' x(1671-1) x 10'"

. 2 1 .

W =169.171 J

F/A .

V=-Mil.

A[

T=aAO

F=VaAA9:, 2 x 10'. x'(0.8 x 10-')" 10-' x 10

F= 160 N .

FL ,e = L,a, T - x = vi;.

6: h = h. (1 + aAO)h = 75(1+ 0.00002 " 27}h = 75.0405 cm

5 .

7 .

4 .

Gain or loss in time due to thermal expansion

1 . . (. 0.000036 /)~.1I='2.xA9xt"a (1- 3 I"~~

~~~~~t~~n ~i:G (j sec~' .\

1 0.000036 \ .~.JiB';'\:. At= -2 x '\' i2..3~.. x'20 "~4...".3600

..... illF .'\ \ . iJI'\At=c10..3681~~"\ Yffff . .4&:!";'" ',....;.,:'. . < tt# ._ .- '. - .

Note: If\we increase temperature then.time period increases and watch becomes slow.

8 .( l ~ \ + ) \ : ' ; ' \ \ . ~V'

. l =l"-x ~ \;'. 0,....... 2

~_ F".i~.. V -

",e

. \ \Fd

e~VA'- = extension or compression due to force

. .

FL2

e = L,a, T + x = Y A : (2) 2

By adding eqmition (1) and equation (2)

We get

Ell'oividif1(!{l) ?nd (2)

.We get.'

8o,

"' , uru (rlpa omp ex, M.P. Nagar~Zone-II, Hhol)al (M.I).), 1>11.:425335514253-155 15


16/35

9 . At = 1.91.mm = At, +At.=:> 0.191 em= t,a,Ae+t,a,Ae.=:) 0.191 =(30 ~17 ~10'" + 70 a,) ~100.

a.=2 ~ 10..0

FY = - ; : ; ; x e F, = F.Y,Aa,Ae;= Y.Aa. Ae.

Y ,a, 1.3x10" x1.7~10-sY ------- = 1.105x 10"N/m':. = ~ = 2x10-5

"';

I I a'

\i / J I c :

---t,---

---1--+

!Illffili

B, =(46 - 30)gm

=:> S, = 15.5gmwt = V. P,9

15.5 . - 1216= (1 + 3a, ~15) ~ 1.24

i f a,>~,

.,\.aUI .C. -If r O c l s are free to expand.

t, =to(1 +a,O)t. =to(1 +a,O)

F/Ay=xlt

Fxix=-. AY

_ fl(15.5~1.24)_1lx~a, - ~ 16 1.2 J 45

Wo =mg =46 9 wIalO = 27. C,W, = 30 gm wt = Wo- B,alO = 42.'C

W. = 30.5 9wt =Wo- B.

B. V,P2.. ~ = V

1P1 ~

1 2 .

11.

10.

2Fx toAY . =io(l +a,AO)-i

..... (i)

FxtoAY = i-io(1 +a,AO)

by solving (i) and (ii)wege!

....:(ii)

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17/35

3. both discs are in series combination

R=R,+R,

_PF_ t_F+ Pc Ie =~p~Fl_1_+_U~F_T]t~F+ ,-P",e,-11_+_u~c~T)~t~eA A A' 'f"""" A.

'" Peue-=---=45te . PFuF .

H F PF IF 3H e = Pele =20

114. (a)

(b)

( c )

(el )

(e) ..

i= 1{1 +2 x 10-5x20}

l:010{1-4x 10-4)

t= 10{ 1+4x10-4}

% it=- 4 X 10-2 %

. -4x10-2

%l2= 1+4x10-4 %;: -4x 10-2%

In the figure-h, = 52.8 em, .h

2= 51 em and h =49 em

Now pressure at B =pressure at c.Therefore .

p. +h, Pw g h PS' g = Po+h2 PS' 9 -.h PRS' g

= = - . PRS' (h, +h ) = PS' (h2+h)

P O '"

(1+.95r) h2 +h'

~ =' h,+h(1+5'() .

1+5y _ 51+49 .100

= = - 1+95'1- 52.8+49 = 101.8

Solving Ihis equation, we gel .\

r= 2 x 10'" per "C.

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.".'. ~. "Y. " - ""1. -.-


18/35


19/35

CPP (Heat Transfer) 6th March 11

, H e a lT ra n S 'e r .

Exercise:#1. . . ..

Section (A) : Thermal conduction In linear conductors at steady stateA 1. A uniform slab ofdimension 1Oemx 10em x 1em is kept be.tweentwo Iie.T,1. . .' .Section (C) : Thermal conductIon througll conductors which have not echleved steady, steta

C 1. A metal rod of cross-sectional area 1.0em2 is being healed atone end.At one lime, the temperature gradient

is 5.0'Clem afcross-sec,lionAand is 2.6 .C/em alcross-section B: Calcullite the rate atwllich thetemperatureis increasing in the part~B 01the rod.The heal capacity 01the partAB = 0.40 Jrc. thermal conductivity ofthe material 01the rod =200.W/m-.C. Neglectany loss of tieat to the atmosph~e. .

SectIon (0) : Radiation, stefen's law and weln's law.

D 1. When q, joules of radiation is incident on a body it reflects and transmIts total ofq, joules. Find the

emissivity olthe body. "

D 2. Ablackbody of surfaCe area 1em' is placed inside an enclosure. The enclosure has a constanttemperature

27"C and the blackbody is maintained at 327.C by heating it electrically. What electric pOweris needed to.

maintain the temperature? " =6.0 x 10" W /rn" .:.j('. . .

D 3. Estimate the temperature atwhich a bodymayappear blue or red.The values of 1._ for these are 5000 and

7500 A respectively. {Given Wein's constant b = 0.3 em K I

cpp~

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20/35

[).4 The tempera\)J re of.!! hot liquid i~a container o{ negligible heat capacity falls at the rate of.3 Klm!n due:tq

heat emissiOn to the surroundings, J ustbefore it begijis to solidifY. l:he temperature.then remains constantfor 30 min, by which lime the liquid h'asall slliidified. Find the ratio of specific heat. capacity .of liquid to.specific latent heat of fusion. .' . .. . ..

Sectlon(E) : Newton's Law.of cooling

E 1.. A liquid cools from ;cioC to S0"6i'k5 minutes. F ind tJ ie lime in which itwill further cool down to 50~C, if its

.surrounding is held.at a constant temperature of 30.C. .

,

d

(D) R +. R2, 2.

.(B) bad absorber is good reflector

(D)bad emitter is good absorber .

(b)

* Marked Questions are having more than one corrett option.

Section' (A):: Thermal conduction In linear conductors "at steady state

A 1. A wall has two layers A and B, each made of different material. Both the layers have the same thickness.

The thermal conductivitY for Ais twiCe that of B. Under steadystate, the temperatureili~ere~e across

~~ ;~~Ie wall is 3S .C. ~~~~;~~ temperature di~;)~~~ across the lay~~ ~~.C F't~~. A 2., Two metal cubes with 3 cm-edges of copperandaluminium are arra4~~' . \\~~} ~()...;

as shown in figure '(Ke" =385 W lm-K , K At=' 209 W /m~K ) ','.' . ,..' :. ''''''. '(a) The total thermal C\lrrent from one reservoir-t61he other is :. 1 0 0 " C ,; ; : M ; a 2 O : C .(A) 1.421


21/35

(0) 0.03 .C/sec(C) .044 'C/sec

C 2*, A hollow and' a solid sphere 01 same material and having identical outer slirface are heated under

. identical conditiori .lotlie same temperature at the same time (toth h~ve same e, iI ) : . .'

(A ) in the beginning both will emit e~ual amount 01 radiation per unit time . .

. (B) in the beginninll both will absorb equal amount of radiation per unit time

(C) both spheres will have same rate offall oftemperature (dT/dt)

(0) both spheres will have equal temperatures at any moment

C 3. A metallic sphere having radius 0.08 m and mass iii=10kg is'heated to a temperature o f227'C andsuspended Inside a box whose walls are at iItemperature of 27*C. The maximum rate at which itstemperature will falUs :-

(Take e" 1, Stefan's'constant (J = 5.8 x 10"Wm" K" and specific heat oithe metal s = 90callkg/degJ = 4.2 J oules/Calorie)(A ) .055 'C /s ec (B ) .066 'C /sec

Section (0) : Newton's L!1w of cooling

0-1._ Which of the law can be understood in terms of Stefan;s'law' - i j(A) W ien's displacement law (B) K irchoffs law A z'(C ) N ewton's law of cooling . (0) P lanck's law I'.~~?;~.

0-2,_ A hot liquid is kepI in a big room. Rate 01cooling of liquid (represented as y) is plotted against its temperature

iilI/~' ....':illi; '.,d'T.W hichOfthefOIlOwingC urvesmayrepresentthep(~ ~ J f' ~

y~ Y i At:; . T~';~ '~i_?(A) Ll-.(~)W' (C) M (D) 'L +

T

.

T~! ~. \~ ~.YV T\;.~i~\~(,;" V

bercis9#2.' ".. .,~... .

1, '"Figure shows a'sleel rdd joined to a brass rod. E ach of the rods has length of 31 cm and area of

""cross-(ection 0.20'cm" ~The junction is maintained at a constant temperature 50~C and the two ends are

maintained at 100'C . Calculate the amount of heat taken out from the cold junction in 10 minutes after the

.~ea.dy stille is reached. The thermal conductivities are K_ =46 W/m-'C and K",." = 109 W/n ...C .~ . 5 0 ' C

.100'C I .steel 1--B-ra-SS--\100'C

2, Consider the situation shown in figure. The frame is made of the same malerial and has a uniform

.cross-sectional area everywhere. II amount of heat flowing per second througti a cross~sectionolthe benl


22/35

4. Four lhin identical rods A B . AC , J 3D and E F made of the, same malerial are joined as

shown. The !ree-end$ C. 0 anili' !Ire l]1i1inl?ined at temperatures t,; T 2 and T~respectively. Assiim,ing that there is no loss of heat to the s urroundings, find the'

temperature at joint E when the steady state is attained.

_ S -J 17.C IO.C 27.C

5 .

6 .

7 .

8 .

9 .

10..

11.

12.

One end of copper rod,of uniform cross-section and of length 1:45 m is in contact with ice at O.C and '

the other end with water at 100'C , Find the position of point along its length where a temperature of

200.C should !J e maintained so that in steady state tlie mass of ice melling is equal to that of steam

produced in The same inlerval oftime [Assume thatttie wholesystem'is l,nsulated from surroundings).

(take l;= 540 cal/gL, '= 80 cal/g) " '

F ind the rate of heat flow through a cross~sec~on T c), Thermal conductivity of

the material of the rod is K

R ~ a _ )_ _ L_ _ ff i:~" T">Te H ~

A hollow spherical 'conducting shell of inner radius R , = 0.25 m an~ 'oH!e~~ir;,

radius R , =0.50 m is placed inside a heat reservoir oft~perature T.~ 1090

"C. The shell is initially filledwiih water at O.C. The thermal conductivitY ofth e

material is k = ~: W/m-K and itshe~~iadty is nJ ig;~I~ind th~ine"-.. '.~

,.A,,/' \,'.\ ' K '.-required to raise the temperature of water ,to-(1OO .C.Take sRBcific heat of ,". '.-, " ','~ V '.-; ; okc\ 'A, --'Ii; f R,

4. '- '" '\' , ,,:)j,~22,water s = 4.2 kJ /kg .Gi, de~sity of water ~;=1000kg/m~, ..J t= "'7

A\, ,;~S \~\ " ' 'A cylindrical roo of lengthJ ' m is.filted beme,en a large ice chamber at O.C and an evacuated chamber

,!ffaii\~ined ~m;c~sshciv;n(~l'~u,re: O nly~m a llportio~s of ~e ~ are inside llie chambers arid the rest, '" ,1 ,5 thermally IO sulated from the surrounding. The cross-,seC tiongOing IOtothe evacuated chamberis blackened

'f_4'~ . 'l'- '-" -" i" -.'. " " '" -~_ "" :.'0"',- ',50 that itcompletely'alisorDs any radiation falling on it. The temperature of the blackened end is 17.C when

-'i" '; ': '\. ' l- _ ,: .,,-'t., '".. .

sle?dy state is reacl1ed!Stefan constanla =6 x 1O"W/m'-l(4. F ind the thermal conductivity of the material> --:" '10


23/35

~t.,_!:I.~;;".'~'~.-.," ., ..t,;

B4"C

Insulating\Nalls

Source. of

Energy

--l t-8cm

i O.C

(D) 31/656

(D) 86 'cand 57 'c

(A) 0.75 min

4 .

.. . ,.

2 .

13._ A metal ballof mass 2 k9.Is'healed by means of a 40 W heater.in.a room at 25'C . The temperature.of the ballbecomeS steady at60'C : (a) Find the rate of loss of heat to the surrounding W hen the' ball is at 60'.C. (b)

Assuming Newton's law.of cooling, calculate the rate of loss Q fheat to the surrounding ~en the ball is at39'C . (c) Assume that the temperature of the ball rises uniformly from 25'C 1039'C in~ minutes. F ind the

totallo~s of heat to thE!surrounding during this period. (d) Calculate the specific heatc;apacitY of the metal.

14...> .. Airietalblock of heat capacity 90 J iC placed in a rOO l!)at 25'C Is heated eleCtrically. The heater Is switched", .','offwhenlhe temperature reaches 35~G.The temperature of the block rises at the rate of2'C/s'justaflerthe

" .heater-is'switched on and falls althe rate of 0.2 'Cis just after the healeris switched off. Assume Newton's

.. law.of cooling ~ hold. (a) F ind the power of the heater. (b) Find the power radiated by the block just after theheater Is switched off. (c) Find the power radiated bythe block when the temperatUre of the block is 30.C. (d)As suming that the power radiated at 30'C respresents the average value in the heating process, find the timefor which the heater W as kflpton. .'

~Wi1t.illt!gmmiQDm_1IISingle choice type ." A t. ..)1. Two identical square rods of metal are welded end to end as shown in figure (a), Assume that 10 cal of

heat flows through the rods in 2 min. N ow the rods are welded as shown,in'figure, (b). The'ti~il~ould

take for 10 cal to flow through the rods now, is . . 6 s f ' "i..'" "

. O'C

, (b) .

. (8) 0~5 mi; ~ (C ) 1.5 min? ) (0) 1 min

~ """'" F, ./ ,v~Three metal rods made of. copper, aluminium and brass, each 20 cm long and 4 em in diameter, are"\ 'l\.. 't'- ~ ~ or-.

placed end to end with aluminium between the other, two. T ne free ends of copper and brass are

maintaine


24/35

O. ,tceWater

Chamber,C

, 'A

1,00, .

Steam

Chamber

keq1pe,r rod\l~iti~J 1}\at iOJ lntemperature 20.C}of non.uniform ~~oss section is placed between a

team chambeh~j)o:c and ice-water chamber at D.C . ,', ,

, 'C onduclingrod

5. A spherical solid black body of radius 'r' tl1d.iates:po~er 'H' a,ndits rate of coC1lingis 'C'. If,density Is,

constant then whlCh,of thefoll~wlngiStari!'iNe: .' " . '", " ,

"," '1' " 1(A)H a::rand i: a::r2 (B) H a::r' and c ",' -' (C}H' c i : rand ca:: 2' (D) H a:: ,r' and c a::r'

,r r

More than one choice type

6. T~ bOdies Aand S'have thernlal emissMlles of0.01 and O~81respectiVely. Thesuitai:e areas of the

two bodies ar~the same. The two b~ies emit total nidi~nt power at the same rate. The Wavelength i.ecorresponding to maximum spectral radiancY in the radiation from'S is shifted from the wavelength

, corresponding to maximum spectral radiancy in the radiation frolil,Aby 1.00 11M . If the temperature ofA 11\5802 K ," , , ' [J EE't4~21

(A) the temp~rature of B is 1934 K " (8) A . = ,1.5 II~ '

(C) the temperature of 81s11604 K: (O) the temperature ,of Sis 2901 K4'7. The solar constant is the amount of heat energy received per second pet unitarea of a perfectly blacll

- ~ ..2IfllIt"

surface placed at a mean distance of the Earth from the S un, in the absence of E arth's,atmosph!lre,

the'surface bei,ng held perpendicular to !he direction of sun's rays. I~v~ue is 13~ri1~ '

If the solar constantfor the earth is's'. The surface temperature of the siln,is TK . T he sun subtends II'

small angle 'e' at the earth. Then correct options is/are.:-:- '~." .....'

(A) s a::T'(B) s a::T' (C) s ,a::e' (D)"s a::e ~8, A heated body emits radiation whjc~ has maximlliTi'ftnslly at frllql,lenc"m' IUhe temperatur~ of thi!

body is doubled: " ,~,' ~ ' ' , ,

(A) the maximum intensity radiationviillbe at frequency 2 v,i, ,

(8) the maximum intensityradiatlE! 'wmbeat frequelil:y'"'';,

(C) the total emitted power tiirincrease'by IIfactor 16 . AlA ,


25/35

. , ' : I . , , , ' . , ' . I ~ ': : . ' , . ' . : t ,- - , -~,:; .:;;. -,. ; -~:~ ~-----------------,0K)

,.

_i'ltiiilMPJlijBti- .Cempreh!!RSion #l' .' 4:.

Figureshowsin'cross sectionawall consistingoffour layerswith ttiermalconductivitle$ Kt;=Q.06WI

mK;K;= 0.04W/r'nKand K4

=0.10 W/mK. The layerthicknesses are II =1.5cm.;'L':'=,2.8.cmand

L4

=3'.S'cm. The temperatureof interfacesis as shownin figure. Energy'transfertlirougnih~ w~lI is

steady. ~

, layer 1 , layer 2, Layer"3. layer '4 , .~: : : . , : 'i K,\ K,' K,\'K.itJ

3 0 .~ iil ;: p O . C.;,.., ~,..*t:;--+:~+-Ll~J!.-

3~,f;i.'{,q'M temperature-ofthe mterfacebelweenlayers3 and4 IS: .

~~ifj~1~~a..'".-,(C)2.C .. (O)O.C4., "he temperature6fthe interface betweenlayers2and3is: .'(A)1;1.C.(6)~ (C)7.2.C . (0)5.4.C .

5. 'If layerthlckness 1 .!2 is:'1.4cm, then its thermalconductivity~ will havevalue (InW/mK) :(A)f;


26/35

9 .

~J ...:$ o o ' ~ '.. .~,~''',,", ".;: .'~'l:.U._"';": '_.'~~ .,' -.. .. ..~~'; ... -.. '. i~.'

SrATf:ME ~T-1:~WO ~oiid'cylindrical ro~sofidenlical size and ~i~erent thermal condu.tliviliesK~ .. "iIK, are connected trI series. Then the eqU ivalenttherl'!1al conductiVity of two rod syst~m IS less than the ~'.

value of thermal conduCtivity 'of either rod. ' . . . , . "

O K ,O K ,}ST ATE ME NT,2 : For two cylindrical rods ofid~ntical size and different therJ l)al CQnductivities K, and K,

respectively connected in series, the equivalent thel;l1lal conductivity K is given by

'fiB In the blanks:(l)"../lt is known that the temperature in the room is +20 'Cwhen the outdoor temperature is -20 'C, and

_ +10 'C when the outdoor temperature is -40 'C, A ssuming Newton's law of cooling to be valid, the

temperallireofth.e radiator in the room is___ ' '

2 1 1-=-+-K K1 K2

(A) Statement-l is True, Statement-2 is True; Statement-2ls a correct explanation for Statement-'

(B) Statement~' is True, Statement-2 is True; Statement.2ls NOT a correct explanation for.Statement-l ,

(C) Statement-' is True, Statement-2 is False ~~.(0) Statement-l is False, Statement-2 is True. I os

. "'" . 'Ii. 'llltST ATE ME NT,1 : As the temperature of the blackbody increases, the'wayelength al~~h the spectral

intensity (E,) is maximum decreases. ,,' /'ffi!?Y _'j~"kf~SJ ATE ME NT-2 : The wavelength at which the spectral intensily'willbe maXimum for a black bodyJ s

proporlionalto th.efourth power of its ~bsolute te~p~,@ l~re.~ 11r~'~-(A) Statement-l IS True, Statement-2ls True; Statement-2ls a correct explanation for Statement-1. ', .~,,,.- -'-. .Ii;:;, - ':


27/35

, -.'. - ,-

(vII. Two m'etal cubes Aaild 8 of same size are arranged a(showPl illfigur,: T I U t extellJ e encts'bf

the combin


28/35

T ,

T.

'0

InslAm

furinance,

L' (.V

T , InsuIaIDr .

Oil out

Three rods maae of the samE !m;lterial and having the same crO sS .:seC nonflavE!':. ~.. '. ee o cjoiried as shown Irithefig.E!lch rod i s of same length. The.left and right ends are ;~;":- .... : .keplal O 'C andSO'C respectively. The temperature ofthe junction ofthethi'ee .rOds O"C . . .will be' : . . .'. . ". [J E E (S er;) 2001,1/35]. 9 O " C . .

(A) 45'C : .' (B) eo'c (C) 30'C " (0)20'CThe temperal~re of bodies X and V vary with time as shown in the figure. If

. emis.sivity of bodies X.arld V are ex&ey and absorptive powers are Ax and A,..

(!issume olher Conditions are Identical for both):then:[J E E (Ser.) 2003, 3I84,-:-1J f\o:

Dt. (A) ey> e x' A y > A . (il) ey < ex' Ay< Ax (C) B y > ex; Ay


29/35

1S;

(4)(r. - r,)

't 'Ifi" - .:.;.,tR E 181 ! S gR~1'''':S ~"':,'-. ,.' '- .~- 'lU::;."%C~ .,;w.u.lV(~ '~8'. '"".. ~,.~..

Infrared radiations are detected by , (AlEEE-2002, 41300)

(1) spectrometer (2) pyrometer (35 nanometer (4) photometerW hich of the following is more close to a black body? . (A IE E :E -2002,4I300)

(1) Black board paint. (2) Green leaves

, (3) Black holes (4) Red roses

Which of the following radiations has the least wavelength? (A1EEE-2003, 41300)

(1)'Y -ray~ " ,,(2)~rays (3) a-rays ' (4) X-rays/;~

If the termpreaiUre of the sun were to increasefroQ'l T to 2T and its radius from R to 2R.th~the,ratio olthe '

radiant energy received on earth to what it was previously will be-At, ,',' ~(Ale~,.,.,E',:2"004, ~13,00)(1)4 (2)16 " ,(3) 32 ' < iF (~)64 . '1 \ ' fU WY :

'. 'v' ," "!~_.AThe temperature of the two outer surfaces of a composite slab. copsistingA4- x - 49! twq m!!teria!S K afld 2 1 < and thickness x and 4x, respectively, and T 2 and .,.T,(T,> T,). The rate of heat transfer through Ihe S la~.;' \ \ ~ ,;e( A ( T - T ) 1


30/35

Answe rs

(. S ) - '. . 4llo{T,'

Required proportJ onalily constant = -K - ~1

(A) 13. (C) 14. 9 15. 9

11.

02. O.73W.

81. 15W/m-'C

A 4.2 :."

Section (~) :

Section (e) :

C 1. 12.C/s.I

,Section (0):

10. (e) 1: 4, (b)2.9 : 1

. KnR,R,(TH- To)6. L

8. 3.6W/m-'C

9. 22W

7 .10

5500ln '91:. (2)' 2. (1) 3. (1) 4. (4)

5. (4) 6. (3) 7. (3) 8. (3)9. (1)

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31/35

p;;:~~';'j..l,;,.,.j., \.,!

6 .

1 1 .

10.

s .

3.

4 .

2.

Mal

--Single choice type '. . .'1, Heat is flowing througli two cylindrical rods made of same materials whose ends are maintained at similar

lemperatures. If diameters of the rods are in'ratio 1 : 2 and lengths in ralio 2 : 1, then the ratio of thermalcurrent through them in steady state is :

(A) 1 : 8 (B) 1 : 4 . (C}1 : 6 . (D) 4: 1Two rods of same dimensions, but made of different materials arejoined 'endto end with their free ends beingmaintained at 1oere and OO Crespectively.The temperature of the junction is 70'C. Then the lemperature ofthe junction If the rods are Inlerchanged will be equal to:(A) 10'C (B) 300C (C) 9O"C (D)40"C

The ends of a metre s tick are maintained a11000Cand O 'C. One end of a rod is maintained aI25C. Whereshould its other end be louched on the metre stick so thaI there Is no heal current in the rod in steady state?(A ) 25 cm from llle hot end (B ) 40 cm frO!'(1the cold end A ' , . . ' " " , ' , . f , t J > " .(C)2Scmfromthecoldend ." . (D l.60cmfromthecoldend f:i . 1 1 . : " , "A calorimeter contains SO g of waler al 50'C . The temperature falls to 4~C in 10minUtes"Wh~thecalorimeter contains 100 g of water at 50'C, ittakes 18 minutes for the IE !mperature10becomeAS'C ,thEmthe waler equivalent of Ihe calorimeter is : (Assume rate of heat transfer'tobeconstant) ~ .~. :;~

~~~~I~C sphere haVin~B~~i~~~.08 m and ma~~)~~10k9 is het1~ ~~f;':~eralur:~i~?~.~'a~suspended inside a box whose walls are at a temperalure of 27\C. Tne maximum rate afwhich its .temperature will fllil is:- . \7 J i :r J i .. , 8 ', , ,~~T:~e2eJ ~~le;;C f~~'rte)onstant G" ~1~ Wm" K,\Cd:~ifiC he~~~e{n~= 90 callkgldeg

(Al .055 'Clsec (8) .0~6~Clsec(C):044 'Clsec.(fND) 0.03 'C/sec

A cylinder of radi~s R made of.a material bf thermal con:luclivity K, is surrounded by a cylindrical shellof inner radius R &n,9.oufer.rad.il.l~2~ made:of a malertal'ofthem1'al condu.ctivity K,. The two ends oflhecombined system are maintained at!Wo dl~rent temperatures. There I S no loss of heal across thecylindrical surface ar,dlhesystem is irf steady state. Th~effective thermal conductivity oflhe system

. ' is: (t~\~!.,\ K \J ,'~~' . [JEE'SS.2]j (~)~,+K,l~\\~)rK l~~~' V(e) K1+:. . (D)3K:;t


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i e ,

1 " A hollow tube has it length I, inner radius R1

ahdouter radius R,. The material has thermalconducliVitYK'Find rate of heat flowing throullh the walls 0!I1etube if the flat ends are maintained at !emperatures T, and-

T.(T . >T,). ' , ,

2. CalcUlate!I1emai conduCtance'for radial flow of an annular cylinder of leng!l1iand inneiand, ' .~.'outer radius r, and r. Assume that !I1ermal conduclivilyolthe material is K. I . I

3. Calculate thermal conductance for radial flow of a spherical shen of inner and o~ter radiUS r, and r. Assume

thattherm'al conductivity olthe materialis K " '

4 . A metallic cylindrical vessel ~ose inner andouter ra~ii are r1

and r,!s fiulid with ice at oO C . The mass oltheice in the cylinder is m. C ircular portions of the cyUnderis sea ed witn'completely adiaba~c walls. The vessel;s kept in air. Temperature of the air is 50"C. F ind time elapsed for the ice to melt completely. (Thermalconductivity of the cylinder is k, its length is i.Latentheat of fusion is L). "

5. A uniform cylinder of length L and thermal conductivity k is placed on 'a metal plate of thesaR1e area 5ofmass m and infinite conduclivity. The ~pecific heat of,the plate is c. The top of the cylinder is maintained at

To. Find the time required for the temperature of the plate to rise from T; to T. (T, '" < . : " "ih .'if '

9. An electric heater'is. used in a room'of totarwall area.j 37 m'to maintain a temperature of 20.C insideit when ou!J ;iqe temp~ratureis -1 O.C\Thewalls have ihree different layers of materials. The innermostlayer is-o'! wood of thickneSs 2.5 cmlJ he'llliddle layer is of cement of thickness 1.0 cm and theo.u,termb~t layer~s of iirt~ko! tllickness


33/35

:> , r~~.' ;

,'- .~~,;:.

,Iti.',:;

T. ', ",. "J~" ~".

lnitialteinpetiilure dlfferel1ceis maxlmum&hence

initial rate fanotiemperature is m~ximum

d e . .. mSdt" ;' 4nrOa (T' _ - T.)

de 4nrza::)-=.--(T' -T' )

dt ms--

...""--., . ",,"',

. .. l

Thermal resistance R =K A '.for same temperature differenCe,thermal

. 1cu.lTenta R.

1 .

, . r. -, . '

j :'i \f tl l tf so l d I I Q " : s.

i,

41


34/35

In (To - T 2) '_ . ks iT o - T 1 - mcl

I" _mcl_ l n l _ T . ~ 0 _ - _ T 2 ~ '1. ksTo - T1

~

.', .. . dr

IdR" f2it

rtk

'.

I t I, dl ks-mc --" -fdt

T,To-Tlo

@.dr';

,.

in r2/r 1 . 1 .Solving R" 211 tk. G = R . .

A"1.6m' e"1

dQdt "P aeA T'" (6 N 10-")(1) 1.6m2(310)' .

= 887 J

mL=e2; O , ) . time.

~ Where L =lalenl heat of fusion .. ,

. mL x R mltn r 2 / r ,Time =(9

2_ 9

1) = 211 lk.50

The whole metal plate wiII.always be at uniform

temperature (T) s ince ii has infiinite conductivity.

dT ks(To- T)Ihen, -mc dt" l

. ". fdR "f dr ..

411 - r2 k"

(rAR"4 ..

.llJ1'r2,k

6 .

5 .

2.

3 .

, .

"~

,"-.;;

C MUMInatJ ir

R" 12em

S"T S"T

p"450W"


35/35

7.i..

8 .

9. '

10,'

(a) e'a AT4=0:55 x 6.x 10-8 x 1.5x (323)4

=53~W .' ':'" "(b), eaAT"s;.,.=O:55x'6x 1()'-8x 15x(295)4

,=375W':, .

(c) 539~~75='164W'

e i s A(T4- T\,ino~ingol =0.97 ,x5,67 x 10-8.2,(3014-29J 4) =92.2W, ,

A=137m2

. 'dQ ~0...+-10) ,IH= dt = R,+R:z+R

3=9kW

-'1O'C B!J eks Cement' Wood 20'C

where

dO Ks11...7t" = T{T, -T2) ,

where T1and T" iire temperatureS of ~'chunksas function of lime 'f.' , '. ' '

~C dT,=Ks (T - T) ,'dl,l' 2

'dT, ,'Ks " ,C'-'-dt = -, (T - T), 'l' 2.

or

or

or

. ( T ) X l tT(x)=T, ...l. '

, T,

1 dOq=--

A dt

Heat Transfer Calorimetry

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